Bayesian Nash Equilibrium

We have already seen that a strategy for a player in agame of incomplete information is a function that speci-fies what action or actions to take in the game, for everypossible type of that player.

A Bayesian Nash Equilibrium is a Nash equilibrium ofthis game (in which the strategy set is the set of actionfunctions).

There are two ways of finding a pure-strategy BayesianNash Equilibrium (BNE).

Method 1. This method works directly on the Bayesiannormal form representation, which is most easily done byconverting the game into the corresponding payoff ma-trix. Simply find the Nash equilibria from the payoff ma-trix.

This method computes expected payoffs from an ex anteperspective, before the players learn their types. Noticethat, if the set of actions available or the set of possibletypes is infinite, we cannot construct the payoff matrixso method 1 will not work.

Here is the payoff matrix for the Entry Game with CostUncertainty, with best responses marked with a star.

firm 2E N

ELEH −14,−14

34∗, 0∗

firm 1 ELNH 0∗, 14∗ 1

2, 0

NLEH −14,14∗ 1

4, 0

NLNH 0∗, 34∗

0, 0

There are three BNE: (ELEH,N), (NLNH,E), and(ELNH,E).

Here is the payoff matrix for the Gift Game, with bestresponses marked with a star for the case in which player1 is more likely to be a friend p > 1

2.

player 2A R

NFNE 0, 0∗ 0∗, 0∗

player 1 NFGE 1− p, p− 1 p− 1, 0∗GFNE p, p∗ −p, 0GFGE 1∗, 2p− 1∗ −1, 0

Thus, when p > 12, the BNE are (N

FNE,R) and (GFGE,A).

Here is the payoff matrix for the Gift Game, with bestresponses marked with a double star for the case in whichplayer 1 is more likely to be an enemy p < 1

2.

player 2A R

NFNE 0, 0∗∗ 0∗∗, 0∗∗

player 1 NFGE 1− p, p− 1 p− 1, 0∗∗GFNE p, p∗∗ −p, 0GFGE 1∗∗, 2p− 1 −1, 0∗∗

Now the game has only one BNE, (NFNE,R). Nomatter what p is, it is an equilibrium for player 1 neverto offer because the gift will be refused. There is also aBNE in which player 2 accepts and both types of player1 offer, but only if type F is more likely than type E.

Method 2. This method for finding the BNE convertsthe game into an equivalent "bigger" game in which thedifferent types of each player are treated as separate play-ers. The payoff to player (i, ti) is the expected payoff ofplayer i, conditional on being type ti.

Any NE of the bigger game is a BNE of the original game,and vice versa:

The payoff in the Bayesian normal-form matrix is thesummation over all types of the probability of a type mul-tiplied by the expected payoff conditional on that type.

=⇒If player i is best responding in the original game, thenit is impossible to increase his payoff conditional on anyof his types (or else this summation would be higher), soeach player (i, ti) must be best responding in the biggergame.

⇐=If each player (i, ti) is best responding in the big-ger game, then it is impossible to increase his ex anteexpected payoff in the original game by choosing a differ-ent strategy function ai(ti). (The only way to achieve ahigher summation of probability of ti multiplied by util-ity conditional on ti is to increase at least one term inthe sum, which is impossible if each player (i, ti) is bestresponding in the bigger game.)

Method 2 is often easier than Method 1, especially whenplayers have an infinite number of possible actions (Cournotgame with cost uncertainty) or an infinite number of types(auction with a continuous distribution of possible valu-ations for the object being auctioned).

Let us apply Method 2 to the Entry Game with CostUncertainty, and see that we get the same answer as inMethod 1.

firm 2E N

firm 1 E −c1,−c2 1− c1, 0N 0, 1− c2 0, 0

In the bigger game, there are three players, (1,H), (1, L),and 2. Let us find the NE of this game.

Can there be a NE in which firm 2 chooses E? Player(1, H) receives a payoff of 0 by choosing NH, and apayoff of −12 from choosing EH, so the best responseis NH. Player (1, L) receives a payoff of 0 by choosingNL, and a payoff of 0 from choosing EL, so both NL

and EL are best responses.

Firm 2’s choice of E is the best response to the pro-file (NL,NH) [because 3

4 > 0] and it is a best re-sponse to the profile (EL,NH) [because 1

4 > 0], soboth (NL,NH,E), and (EL,NH,E) are NE of the 3-player game. Therefore, (NLNH,E), and (ELNH,E)

are BNE of the Bayesian normal-form game.

Can there be a NE in which firm 2 chooses N? Player(1, H) receives a payoff of 0 by choosing NH, and apayoff of 12 from choosing EH, so the best response isEH. Player (1, L) receives a payoff of 0 by choosingNL, and a payoff of 1 from choosing EL, so the bestresponse is EL.

Firm 2’s best response to the profile (EL,EH) is N[because 0 > −14]. Therefore, (E

L,EH,N) is a NEof the 3-player game, and (ELEH,N) is a BNE of theBayesian normal-form game.

Cournot Competition with Cost Uncertainty

Consider a (simultaneous move) Cournot game with theinverse demand function

p = 1− q1 − q2.

Firm 1’s production cost is zero.

Firm 2 has two possible types, L and H, each of whichoccur with probability 12.

A type L firm 2 has low marginal cost, 0, and a type Hfirm 2 has high marginal cost, 14.

In the Bayesian normal-form game, a strategy for firm 1is a quantity, q1, and a strategy for firm 2 is a functionthat specifies a quantity for each type, (qL2 , q

H2 ).

Since every nonnegative quantity is a possible strategy forfirm 1 and every pair of nonnegative quantities is a possi-ble strategy for firm 2, the resulting payoff matrix wouldhave an infinite number of rows and columns! ClearlyMethod 1 will not be easy.

Under Method 2, we consider the three player game withfirm 1, firm 2L, and firm 2H. To find the NE, we com-pute the best response functions for all three players andsolve the three equations for the three NE quantities.

Starting with firm 2L, its payoff function is

uL2 = (1− q1 − qL2 )qL2 .

Differentiating with respect to qL2 , setting the expressionequal to zero, and solving for qL2 , we can solve for firm2L’s best response function.

∂uL2∂qL2

= 0 = 1− q1 − 2qL2

BRL2 (q1) =

1− q12

.

The payoff function of firm 2H is given by

uH2 = (1− q1 − qH2 )qH2 −

qH24.

Differentiating with respect to qH2 , setting the expressionequal to zero, and solving for qH2 , we can solve for firm2H’s best response function.

∂uH2∂qH2

= 0 = 1− q1 − 2qH2 −1

4

BRH2 (q1) =

3

8− q12.

The payoff function for firm 1 is based on the expectationthat half of the time it is competing with firm 2L andhalf of the time it is competing with firm 2H.

u1 =1

2[(1− q1 − qL2 )q1] +

1

2[(1− q1 − qH2 )q1]

= (1− q1 −qL22− qH2

2)q1

Differentiating with respect to q1, setting the expressionequal to zero, and solving for q1, we can solve for firm1’s best response function.

∂u1∂q1

= 0 = 1− 2q1 −qL22− qH2

2

BR1(qL2 , q

H2 ) =

1

2− qL24− qH2

4.

The Nash equilibrium is the solution to the following threeequations

BRL2 (q1) =

1

2− q12= qL2

BRH2 (q1) =

3

8− q12= qH2

BR1(qL2 , q

H2 ) =

1

2− qL24− qH2

4= q1

To solve, substitute qL2 from the first equation and qH2from the second equation into the third equation, andsolve for q1.

1

2− (1

8− q18)− ( 3

32− q18) = q1

9

32+q14

= q1

q1 =4

3(9

32) =

3

8.

Substituting q1 =38 into the remaining two equations, we

have the Nash equilibrium strategy profile for this threeplayer game, q1 =

38, q

L2 =

516, q

H2 = 3

16.

Thus, the BNE for the original game with two players isthe following:

q1 =3

8

(qL2 , qH2 ) = (

5

16,3

16).

For this game, firm 1’s quantity is a best response to theaverage quantity selected by firm 2.

Auction Markets

We will solve the games corresponding to various auctionrules under the following environment:

There is one indivisible object being sold.

The players are the n bidders. Each player has a valuationfor the object, with player i’s valuation denoted by vi.

If player i wins the auction and makes a payment, p, heroverall payoff is vi − p; if she does not win the auctionbut she makes a payment, p, her overall payoff is −p.

We assume that each vi is independently drawn from theuniform distribution over the unit interval [0, 1]. In otherwords, all realizations between 0 and 1 are equally likely,and knowing vi provides no information about the otherplayers’ valuations.

1. Sealed-bid, first-price auction.

The players simultaneously submit bids, where the bidof player i is denoted by bi. Each player observes hervaluation (her type) before deciding what to bid, so astrategy is a bid function, bi(vi).

The player submitting the highest bid wins the auctionand makes a payment equal to her bid.

Players who do not win the auction do not make a pay-ment; their payoff is zero. In case of a tie for the highestbid, someone is randomly selected as the winner.

What would you bid if n = 10 and your valuation is 0.6?

Notice that it does not make sense to bid more than yourvaluation, because your payoff cannot be positive. Eitheryou lose the auction and receive zero, or win the auctionand receive a negative payoff.

In fact, players should bid less than their valuation so thatif they win the auction, their payoff is positive.

Finding the BNE is not easy. We will guess that thereis a symmetric BNE in which all players bid a constantfraction of their valuation:

bi(vi) = avi

for some number a that is the same for all players and isbetween zero and one.

Then we will use the condition that every type of everyplayer is best-responding to the other players by biddingthis way. This will allow us to solve for the value of athat make this symmetric profile of bidding functions aBNE.

Consider player j with valuation vj, and suppose that allof the other players are bidding according to bi(vi) = avi.Then the highest possible bid by one of the other players(with vi = 1) is a, so there is no reason to bid more thana.

If player j makes a bid of b, she wins the auction if andonly if all of the other bids are below b. Based on theirbidding functions, this happens if we have for all i 6= j,

avi < b, or

vi <b

a.

Because of the uniform distibution, the probability that aparticular one of the other players has a valuation belowba is

ba.

Then the probability that all of the other players havevaluations below b

a, so that player j wins the auction, isgiven by

pr(j wins when bidding b) =µb

a

¶n−1.

This allows us to express player j’s payoff as a functionof her valuation and her bid, given the bidding functionsof the other players.

uj =µb

a

¶n−1[vj − b]

We can now find the optimal bid for each type of playerj, by taking the derivative of uj with respect to b, settingthe expression equal to zero, and solving for b.

a1−n[(n− 1)bn−2(vj − b) + bn−1(−1)] = 0

(n− 1)(vj − b)− b = 0

(n− 1)vj − nb = 0

b =n− 1n

vj

Thus, if other players are bidding a constant fraction oftheir valuations (no matter what the constant, a), thebest response of bidder j is to bid a constant fraction,n−1n , of her valuation.

Therefore, we have a BNE if each player i uses the biddingfunction,

bi(vi) =n− 1n

vi.

Recapping, in the sealed-bid first-price auction, the BNEis for all players to use the bidding function, bi(vi) =n−1n vi.

Players shade their bid below their valuation, to balancethe profit when they win against the risk of not winning.The more players in the auction, the less they can affordto shade their bid.

The BNE outcome is efficient, because the player withthe highest valuation always wins the auction.

When n = 2, players bid half their valuation, so whenthe highest valuation is v, the player with that valuationreceives a payoff of v2 and the seller receives revenue ofv2.

2. Sealed-bid, second-price auction.

The players simultaneously submit bids, where the bidof player i is denoted by bi. Each player observes hervaluation (her type) before deciding what to bid, so astrategy is a bid function, bi(vi).

The player submitting the highest bid wins the auctionand makes a payment equal to the second highest bid.

Players who do not win the auction do not make a pay-ment; their payoff is zero. In case of a tie for the highestbid, someone is randomly selected as the winner.

For example, if player 3 submits the highest bid of 0.78and player 6 submits the second-highest bid of 0.62, thenplayer 3 wins the auction (driving home with the object)and pays 0.62, while the other players do not receive theobject or make any payments.

What would you bid if n = 10 and your valuation is 0.6?

One could try to solve for the BNE of the sealed-bidsecond-price auction the same way that we solved thefirst-price auction, but there is a much easier way.

Notice that it is a weakly dominant strategy to bid yourvaluation:

If the highest of the other players’ bids (call it v0) isgreater than vi, then bidding vi is a best response. Playeri loses, but changing her bid in order to win would requireher to bid more than v0, in which case she would pay v0,which is more than her valuation.

If v0 < vi, then bidding vi is a best response. Player iwins the auction and receives a positive payoff. Changingher bid while still winning does not change her payment,v0, and changing her bid to something below v0 reducesher payoff to zero.

The symmetric BNE has all players choosing the biddingstrategy bi(vi) = vi.

The BNE outcome is efficient, because the player withthe highest valuation always wins the auction.

When n = 2, since players bid their valuation, whenthe highest valuation is v, the player with that valuationwins the auction and makes a payment equal to the otherplayer’s valuation. The payment is uniformly distributedover the interval from 0 to v, so the expected paymentis v2. Thus, the winner receives an expected payoff of

v2

and the seller receives expected revenue of v2.

The expected payoff to the players and the expected rev-enue to the seller is the same as in the first-price auction.This result that payoffs do not depend on the auctionformat (called revenue equivalence) extends to n playersand to valuation distributions other than uniform.

These sealed-bid auctions have dynamic counterparts.

The (independent private values) second-price sealed bidauction is essentially equivalent to the ascending-price orEnglish auction you see in movies or on eBay.

The (independent private values) first-price sealed bidauction is essentially equivalent to the descending-priceor Dutch auction.

The game changes significantly if payoffs have a "com-mon value" component. That is, if the object’s worth tome increases when I learn that the object is worth a lotto you.

For example, if we are auctioning two tickets to the Buck-eye game on the 50 yard line in the first row of C-deck,then the bidders know their valuations, which would notchange if they learned that the other bidders had highvaluations. If instead the seat locations are not specifiedand bidders’ types in part reflect information about wherethe seats are, then learning your type tells me somethingabout what the seats would be worth to me.